Parameter converting method for bifurcation analysis of nonlinear dynamical systems

Document Type : Article

Authors

Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, P.O. Box 484, Postal Code 47148-71167, Mazandaran, Iran.

Abstract

For detecting behavior of a dynamical system, bifurcation analysis is necessary with respect to change in parameters of system. In this work, based on the solution of ordinary differential equations from initial value and parameters, a simple method is presented, which can efficiently reveal different bifurcations of system. In addition to its simplicity, this method does not required to have deep physical and mathematical understanding of the problem, and because of its high precision and the speed of solutions, does not need to reduce the order of models in many complex problems or problems with high degrees of freedom. This method is named parameter converting method (PCM), which has two steps. In the first step the parameter is varied as a function of time and in the second step, time is expressed as inverse of this assumed function. With this method bifurcation and amplitude-frequency diagrams and hidden attractors of some complex dynamics will be analyzed and the sensitivity of the multi potential well systems to initial conditions is studied. With this algorithm, a simple way to find the domain of high-energy orbit in bistable systems is obtained.

Keywords

Main Subjects


References:
1. Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley Classics Library Edition Published (1993). 
2. Pakdemirli, M., Karahan, M.M.F., and Boyac, H."A new perturbation algorithm with better convergence properties: Multiple scales lindstedt poincare method", Math. Comp. App. 14, pp. 31-44 (2009).
3. Pakdemirli, M. "Review of the new perturbationiteration method", Math. Comp. App., 18, pp. 139- 151 (2013).
4. Damil, N. and Potier-Ferry, M. "A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures", Int. J. Eng. Sci., 28, pp. 943-957 (1990).
5. Vannucci, P., Cochelin, B., Damil, N., and Potier- Ferry, M. "An asymptotic-numerical method to compute bifurcating branches", Int. J. Num. Meth. Eng,. 41, pp. 1365-1389 (1998).
6. Boutyour, E.H., Zahrouni, H., Potier-Ferry, M., and Boudi, M. "Bifurcation points and bifurcated branches by an asymptotic numerical method and Pade approximants", Int. J. Num. Meth. Eng, 60, pp. 1987-2012 (2004).
7. Cochelin, B., Damil, N., and Potier-Ferry, M. "Asymptotic numerical method and pade approximants for nonlinear elastic structures", Int. J. Num. Meth. Eng., 37, pp. 1187-1213 (1994).
8. Hamdaoui, A., Hihi, R., Braikat, B., Tounsi, N., and Damil, N. "A new class of vector pade approximants in the asymptotic numerical method: Application in nonlinear 2D elasticity", World J. Mech., 4, pp. 44-53 (2014).
9. Elhage-Hussein, A., Potier-Ferry, M., and Damil, N. "A numerical continuation method based on Pade approximants", Int. J. Sol. Struc., 37, pp. 6981-7001 (2000).
10. Cochelin, B. "A path-following technique via an asymptotic-numerical method", Comp. Struc., 53, pp. 1181-192 (1994).
11. Kerschen, G., Peeters, M., Golinval, J.C., and Vakakis, A.F. "Nonlinear normal modes, Part I: A useful framework for the structural dynamics", Mech. Sys. Sig. Proc. 23, pp. 170-194 (2009).
12. Peeters, M., Viguie, R., Se randour, G., Kerschen, G., and Golinval, J.C. "Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques", Mech. Sys. Sig. Proc., 23, pp. 195-216 (2009).
13. Barton, D.A.W., Krauskopf, B., and Wilson, R.E. "Homoclinic bifurcations in a neutral delay model of a transmission line oscillator", Nonlinearity, 20, pp. 809-829 (2007).
14. Beyn, W.J. "The numerical computation of connecting orbits in dynamical systems", IMA J. Num. Ana., 10, pp. 379-405 (1990).
15. Beyn, W.J. "Numerical methods for dynamical systems", W. Light, Ed., Advances in Numerical Analysis, Lancaster, Clarendon, Oxford, I, pp. 175-236 (1990).
16. Keller, H.B., Lectures on Numerical Methods in Bifurcation Problems, Tata Institute of Fundamental Research (1986).
17. Allgower, E.L. and Georg, K., Introduction to Numerical Continuation Methods, Springer Series in Computational Mathematics (1990).
18. Meijer, H., Dercole, F., and Oldman, B. "Numerical bifurcation analysis", Eds., R., Meyers, Mathematics of Complexity and Dynamical Systems, Springer, New York, NY (2012).
19. Doedel, E.J., Nonlinear Numerics, J., Franklin Inst., 334(5-6), pp. 1049-1073 (1997). 
20. Karkar, S., Cochelin, B., and Vergez, Ch. "A highorder, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities", J. Sound Vib., 332, pp. 968-977 (2013).
21. Karkar, S., Cochelin, B., and Vergez, Ch. "A comparative study of the harmonic balance method and the orthogonal collocation method on stiff: nonlinear systems", J. Sound Vib., 333, pp. 2554-2567 (2014).
22. Rega, G. and Troger, H. "Dimension reduction of dynamical systems: methods, models, applications", Nonlinear Dyn, 41, pp. 1-15 (2005).
23. Steindl, A. and Troger, H. "Methods for dimension reduction and their application in nonlinear dyn", Int. J. Sol. Struc., 38, pp. 2131-2147 (2001).
24. Terragni, F. and Vega, J.M. "On the use of POD-based ROMs to analyze bifurcations in some dissipative systems", Phys. D., 241, pp. 1393-1405 (2012). 
25. Couplet, M., Basdevant, C., and Sagaut, P. "Calibrated reduced-order POD- Galerkin system for fluid flow modeling", J. Comp. Phys., 207, pp. 192-220 (2005). 
26. Sirisup, S., Karniadakis, G.E., and Kevrekidis, I.G. "Equations-free/Galerkin-free POD assisted computation of incompressible  ows", J. Comp. Phys., 207, pp. 568-587 (2005).
27. Rapun, M.L. and Vega, J.M. "Reduced order models based on local POD plus Galerkin projection", J. Comp. Phys., 229, pp. 3046-3063 (2010). 
28. Terragni, F., Valero, E., and Vega, J.M. "Local POD plus Galerkin projection in the unsteady lid-driven cavity problem", J. Sci. Comp. arch., 33, pp. 3538- 3561 (2011).
29. Doedel Laurette, E., Tuckerman, S., Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Springer, The IMA Volumes in Mathematics and Its Applications, 119 (1999).
30. Govaerts, W. "Numerical bifurcation analysis for ODEs", J. Comp. App. Math., 125, pp. 57-68 (2000). 
31. Heyman, J., Girault, G., Guevel, Y., Allery, C., Hamdouni, A., and Cadou, J.M. "Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method", Comp. Fluids, 76,pp. 73-85 (2013).
32. Meyers, R.A., Mathematics of Complexity and Dynamical Systems, Springer Science + Business Media, LLC (2012).
33. Dijkstra, H.A., Wubs, F.W., Cliff:e, A.K., et al. "Numerical bifurcation methods and their application to fluid dynamics: Analysis beyond simulation", Comm. Comp. Phy., 15, pp. 1-45 (2014).
34. Gai, G. and Timme, S. "Nonlinear reduced-order modelling for limit-cycle oscillation analysis", Nonlinear Dyn, 84, pp. 991-1009 (2016).
35. Feng, Y. and Pan, W. "Hidden attractors without equilibrium and adaptive reduced-order function projective synchronization from hyper chaotic Rikitake system", Pramana J. Phy., 88, p. 62 (2017).
36. Chnafa, C., Valen-Sendstad, K., Brina, O., Pereira, V.M., and Steinman, D.A. "Improved reduced-order modelling of cerebrovascular  ow distribution by accounting for arterial bifurcation pressure drops", J. Biom, 51, pp. 83-88 (2017).
37. Amabili, M., Karazis, K., and Khorshidi, K. "Nonlinear vibrations of rectangular laminated composite plates with different boundary conditions", Int. J. Struc. Stab. Dyn., 11, pp. 673-695 (2011).
38. Kurt, E., Ciylan, B., Taskan, O.O., and Kurt. H.H. "Bifurcation analysis of a resistor-double inductor and double diode circuit and a comparison with a resistorinductor- diode circuit in phase space and parametrical responses", Scientia Iranica, Trans. D: Computer Sci. Eng. Ele. 21, pp. 935-944 (2014).
39. Sayyaadi, H., Tadayon, M.A., and Eftekharian, A.A. "Micro resonator nonlinear dynamics considering intrinsic properties", Scientia Iranica, Trans. B: Mech. Eng., 16, pp. 121-129 (2009).
40. http://manlab.lma.cnrs-mrs.fr/. 
41. https://sourceforge.net/p/cocotools/wiki/Home/. 
42. http://www.ni.gsu.edu/rclewley/PyDSTool/ Front-Page.html.
43. http://indy.cs.concordia.ca/auto/#documentation. 
44. Lorenz, E.N. "Deterministic nonperiodic  flow", J. Atm. Sci., 20, pp. 130-141 (1963).
45. Rossler, O.E. "An equation for continuous Chaos", Physics Letters, 57A, pp. 397-398 (1976).
46. Chua, L.O. and Lin, G.N. "Canonical realization of Chua's circuit family", IEEE Tran. Circuits Sys., 37, pp. 885-902 (1990).
47. Chen, G. and Ueta, T. "Yet another chaotic attractor", Int. J. Bifu. Chaos, 9, pp. 1465-1466 (1999). 
48. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., and Prasad, A. "Hidden attractors in dynamical systems", Phy. Rep., 637, pp. 1-50 (2016).
49. Kuznetsov, N.V., Leonov, G.A., and Vagaitsev, V.I. "Analytical-numerical method for attractor localization of generalized Chua's system", IFAC Proc., 43, pp. 29-33 (2010).
50. Leonov, G.A., Vagaitsev, V.I., and Kuznetsov, N.V."Algorithm for localizing Chua attractors based on the harmonic linearization method", Doklady Math., 433, pp. 323-326 (2010).
51. Leonov, G.A., Kuznetsov, N.V., and Vagaitsev, V.I. "Localization of hidden Chua's attractors", Phy. Let. A, 375, pp. 2230-2233 (2011).
52. Leonov, G.A. and Kuznetsov, N.V. "Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems", Doklady Math., 84(1), 475-481 (2011).
53. Leonov, G.A. and Kuznetsov, N.V. "Analyticalnumerical methods for investigation of hidden oscillations in nonlinear control systems", IFAC Proc, 44(1), pp. 2494-2505 (2011).
54. Bragin, V.O., Vagaitsev, V.I., Kuznetsov, N.V., and Leonov, G.A. "Algorithms for Finding hidden oscillations in nonlinear systems. The Aizerman and Kalman Conjectures and Chua's circuits", J. Computer Sys. Sci. Int., 50(4), pp. 511-543 (2011).
55. Kuznetsov, N., Kuznetsova, O., Leonov, G., and Vagaitsev, V. "Analytical-numerical localization of hidden attractor in electrical Chua's circuit", Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, 174, Springer, Berlin, Heidelberg, pp. 149-157 (2013).
56. Zelinka, I. "Evolutionary identiFication of hidden chaotic attractors", Eng. App. Art. Intel., 50, pp. 159- 167 (2016).
58. Li, Q., Zeng, H., and Yang, X.S. "On hidden twin attractors and bifurcation in the Chua's circuit", Nonlinear Dyn, 77(1-2), pp. 255-266 (2014).
59. Griff:ths, D.F. and Higham, D.J. Numerical Methods for Ordinary Diff:erential Equations: Initial Value Problems, Springer-Verlag London (2010).
60. Dormand, J.R. and Prince, P.J. "A family of embedded Runge-Kutta formula", J. Comp. App. Math., 6, pp. 19-26 (1980).
61. http://www.scholarpedia.org/article/Rossler attractor. 
62. Kovacic, I. and Brennan, M.J., The Duff:ng Equation: Nonlinear Oscillators and Their Behaviour, Wiley (2011).
63. http://www.scholarpedia.org/article/Duff:ng oscillator. 
64. Harne, R.L. and Wang, K.W. "A review of the recent research on vibration energy harvesting via bistable systems", Smart Mat. Struc., 22(2), 023001 (2013).
65. Panyam, M. and Ram, M., "Characterizing the effective bandwidth of nonlinear vibratory energy harvesters possessing multiple stable equilibria", PhD Dissertation, Clemson University, December (2015).
66. Ramlan, R., Brennan, M.J., Mace, B.R., and Kovacic, I. "Potential beneFits of a non-linear stiff:ness in an energy harvesting device", Nonlinear Dyn, 59(4), pp. 545-558 (2010).
67. Erturk, A. and Inman, D.J. "Broadband piezoelectric power generation on high-energy orbits of the bistable Duff:ng oscillator with electromechanical coupling", J. Sound Vib., 330, pp. 2339-2353 (2011).
68. Karami, M.A. and Inman, D.J. "Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems", J. Sound Vib., 330, pp. 5583-5597 (2011).
69. Panyamn, M., Masana, R., and Daqaq, M.F. "On approximating the eff:ective bandwidth of bi-stable energy harvesters", Int. J. Nonl. Mech., 67, pp. 153- 163 (2014).
70. Panyam, M. and Daqaq, M.F. "A comparative performance analysis of electrically optimized nonlinear energy harvesters", J. Intel. Mate. Struc., 27(4), pp. 537-548 (2016).
71. Guang-Qing, W. and Wei-Hsin, L. "A strategy for magnifying vibration in high-energy orbits of a bistable oscillator at low excitation levels", Chinese Phy. Let., 32(6), 068503 (2015).
72. Zhou, Sh., Cao, J., Inman, D.J., Liu, Sh., Wang, W., and Lin, J. "Impact-induced high-energy orbits of nonlinear energy harvesters", Ap. Phy. Let., 106,093901 (2015).
73. Wang, G.Q. and Liao, W.H. "A bistable piezoelectric oscillator with an elastic magniFier for energy harvesting enhancement", J. Intel. Mate. Struc., 28(3), pp. 392-407 (2017).
74. Zhou, Sh., Cao, J., Inman, D.J., Lin, J., and Li, D. "Harmonic balance analysis of nonlinear tristable energy harvesters for performance enhancement", J. Sound Vib., 373(7), pp. 223-235 (2016).
Volume 27, Issue 1
Transactions on Mechanical Engineering (B)
January and February 2020
Pages 310-329
  • Receive Date: 03 April 2018
  • Revise Date: 13 July 2018
  • Accept Date: 22 October 2018